Convex Hulls Under Uncertainty

Agarwal, Pankaj K.; Har-Peled, Sariel; Suri, Subhash; Yıldız, Hakan; Zhang, Wuzhou

doi:10.1007/s00453-016-0195-y

Cited by 21 publications

(24 citation statements)

References 32 publications

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“…In the first technique, we first identify a set H of points, called the stoch-core of the problem, such that (1): with high probability, all nodes realize in H and (2): conditioning on event (1), the variance is small. Then, we choose Y to be the number of nodes realized to points not in H. We compute the (1 ± ǫ)-estimates for Y = 0, 1 using Monte Carlo by (1) and (2). The problematic part is when Y is large, i.e., many nodes realize to points outside H. Even though the probability of such events is very small, the value of X under such events may be considerably large, thus contributing nontrivially.…”

Section: Perfect Matchingmentioning

confidence: 99%

“…The most probable k-nearest neighbor problem and its variants have attracted a lot of attentions in the database community (See e.g., [11]). Several other problems have also been considered recently, such as computing the expected volume of a set of probabilistic rectangles in a Euclidean space [43], convex hulls [2], skylines (Pareto curves) over probabilistic points [1,7], and shape fitting [32].…”

Section: Related Workmentioning

confidence: 99%

“…In this paper, we study two stochastic geometry models, the locational uncertainty model and the existential uncertainty model, both of which have been studied extensively in recent years (see e.g., [2,3,4,7,25,26,29,30,31], some of which will be discussed in the related work section). In fact, a special case of the locational uncertainty model where all points follow the same distribution is a classic topic in stochastic geometry literature (see e.g., [8,9,10,27,37]).…”

Section: Introductionmentioning

confidence: 99%

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Approximating the Expected Values for Combinatorial Optimization Problems over Stochastic Points

Huang

2015

Automata, Languages, and Programming

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We consider the stochastic geometry model where the location of each node is a random point in a given metric space, or the existence of each node is uncertain. We study the problems of computing the expected lengths of several combinatorial or geometric optimization problems over stochastic points, including closest pair, minimum spanning tree, k-clustering, minimum perfect matching, and minimum cycle cover. We also consider the problem of estimating the probability that the length of closest pair, or the diameter, is at most, or at least, a given threshold. Most of the above problems are known to be #P-hard. We obtain FPRAS (Fully Polynomial Randomized Approximation Scheme) for most of them in both the existential and locational uncertainty models. Our result for stochastic minimum spanning trees in the locational uncertain model improves upon the previously known constant factor approximation algorithm. Our results for other problems are the first known to the best of our knowledge.

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Section: Perfect Matchingmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Approximating the Expected Values for Combinatorial Optimization Problems over Stochastic Points

Huang

2015

Automata, Languages, and Programming

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“…For more details on methods and algorithms in this model see [23]. Computing the convex hull of a set of imprecise points when location of points are given by a probability function is studied in [28], [29].…”

Section: Related Workmentioning

confidence: 99%

The convex hull of finitely generable subsets and its predicate transformer

Davari

Edalat

Lieutier

2019

2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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We consider the domain of non-empty convex and compact subsets of a finite dimensional Euclidean space to represent partial or imprecise points in Computational Geometry. The convex hull map on such imprecise points is given domain-theoretically by an inner and an outer convex hull. We provide a practical algorithm to compute the inner convex hull when there are a finite number of convex polytopes as partial points. A notion of pre-inner support function is introduced, whose convex hull gives the support function of the inner convex hull in a general setting. We then show that the convex hull map is Scott continuous and can be extended to finitely generable subsets, represented by the Plotkin power domain of the underlying domain. This in particular allows us to compute the convex hull of attractors of iterated function systems in fractal geometry. Finally, we derive a program logic for the convex hull map in the sense of the weakest precondition for a given post-condition.

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“…In the multipoint model the problem is NP-hard even for d = 2. Agarwal et al [1] proposed exact and approximation algorithms to compute the probability of a query point lying inside the convex hull of the input.…”

Section: Introductionmentioning

confidence: 99%

Separability of Imprecise Points

Berg

Mehrabi

Sheikhi

2014

Lecture Notes in Computer Science

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Convex Hulls Under Uncertainty

Cited by 21 publications

References 32 publications

Approximating the Expected Values for Combinatorial Optimization Problems over Stochastic Points

Approximating the Expected Values for Combinatorial Optimization Problems over Stochastic Points

The convex hull of finitely generable subsets and its predicate transformer

Separability of Imprecise Points

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